Argument for mathematical independence of size diversity versus zooplankton
biomass and mean zooplankton body size
Mathematically, the zooplankton size diversity calculated with no standardization in
this manuscript is independent of zooplankton biomass and mean zooplankton body
size. First, as can be seen in Figure S5 in the Supporting Information, the three
hypothetical datasets have very different biomasses (integral below the black line),
but have identical size diversity based on the method of calculating size diversity in
this manuscript, indicating that there is no necessary mathematical correlation
between size diversity and zooplankton total biomass. Second, for the two
hypothetical size communities in Figure S6 in the Supporting Information (supposing
that the zooplankton size community was shifted by a real constant a defined as Y= X
– a), theoretically, the calculated size diversity of community Y is equal to that of the
community X, indicating that there is no necessary mathematical correlation between
size diversity and mean zooplankton body size. The mathematical derivation could be
found in the Web Appendix B in Quintana et al. (2008). Thus, in the real ecosystem, if
a "statistical" relationship between size diversity versus zooplankton biomass or mean
zooplankton body size is found, it has biological interpretation but is not caused by
mathematical artifact.
In this research, we did not follow Quintana’s (2008) suggestion to calculate the size
diversity after data standardization by division of sample geometric mean because the
standardization may cause the spurious correlation between the calculated
zooplankton size diversity and zooplankton-phytoplankton biomass ratio. Specifically,
suppose that the body size is standardized by division of sample geometric mean g, as
Y=X/g, the size diversity for Y is:
 (Y )   ( X )  ln g
Eq.1
For the two size communities in Figure S6 in the Supporting Information, the
calculated size diversity after the standardization for community a is μ(X0) –ln15,
which is mathematically different from that of the community b (μ(X0) –ln25).
However, we can intuitively understand that the community a and b should have the
same size diversity.
More importantly, Eq. 1 indicates that the value of size diversity calculated using the
standardization procedure is necessarily determined both by the original size
distribution μ(X) and the geometric mean g. In the East China Sea, the sample
geometric mean of body size (g) shows a significant positive correlation with total
zooplankton biomass (Fig. S7, Supporting Information). Therefore, the
standardization by division of sample geometric mean suggested by Quintana et al.
(2008) may bring the spurious correlation between zooplankton size diversity and
zooplankton-phytoplankton biomass ratio in our research.
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References:
Quintana, X.D., S. Brucet, D. Boix, R. Lopez-Flores, S. Gascon, A. Badosa, J. Sala, R.
Moreno-Amich, and J.J. Egozue. 2008. A nonparametric method for the
measurement of size diversity with emphasis on data standardization. Limnol.
Oceanogr.: Methods, 6:75-86.
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